Apparatus and method for estimating charge rate of secondary cell

ABSTRACT

In charge rate estimating apparatus and method for a secondary cell, a current flowing through the secondary cell is measured, a voltage across terminals of the secondary cell is measured, an adaptive digital filtering is carried out using a cell model in a continuous time series shown in an equation (1), all of parameters at one time are estimated, the parameters corresponding to an open-circuit voltage which is an offset term of the equation (1) and coefficients of A(s), B(s), and C(s) which are transient terms, and, the charge rate is estimated from a relationship between a previously derived open-circuit voltage V 0  and the charge rate SOC using the open-circuit voltage V 0 , 
     
       
         
           
             
               
                 
                   
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             wherein s denotes a Laplace transform operator, A(s), B(s), and C(s) denote poly-nominal functions of s.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to apparatus and method for estimating acharge rate (abbreviated as SOC) of a secondary cell.

2. Description of the Related Art

Japanese Patent Application First Publications No. 2000-323183 publishedon Nov. 24, 2000 No. 2000-268886 published on Sep. 29, 2000, and aJapanese Paper titled “Estimation of Open Voltage and Residual Valuesfor Pb Battery by Adaptive Digital Filter” announced by a JapaneseElectrical Engineering Society (T.IEEE Japan), Volume 112-C, No. 4,published on 1992 exemplify previously proposed SOC estimating apparatusfor the secondary cell. That is to say, since the charge rate (or calledState Of Charge, i.e., SOC) of the secondary cell has a correlation toan open-circuit voltage V₀ (cell terminal voltage when its power supplyof the cell is turned off, also called electromotive force or openvoltage), the charge rate can be estimated when open voltage V₀ isobtained. However, a considerable time is needed until the terminalvoltage is stabilized after the power supply is turned off(charge-and-discharge is ended). Hence, a predetermined time duration isneeded from a time at which the charge-and-discharge is ended todetermine an accurate open-circuit voltage V₀. Therefore, sinceimmediately after or during the charge/discharge time orcharge-and-discharge, it is impossible to determine an accurateopen-circuit voltage and the charge rate cannot be obtained using theabove-described method. Nevertheless, to determine the open-circuitvoltage V₀, the open-circuit voltage V₀ is estimated using a methoddisclosed in the above-described Japanese Patent Application FirstPublication No. 2000-323183.

SUMMARY OF THE INVENTION

However, in the above-described method disclosed in the Japanese PatentApplication Publication No. 2000-323183, open-circuit voltage V₀ iscalculated from a non-recursive (non-regression type) cell model (amodel whose output value is determined only from a present value andpast value of an input value) whose characteristic is wholly differentfrom a physical characteristic of the cell for which an adaptive digitalfilter (sequential type model parameter identification algorithm) isused. The charge rate (SOC) is used from this value. Hence, in a casewhere this method is applied to the actual cell characteristic (input:current, output: voltage), according to the cell characteristic, anestimation calculation is wholly converged or does not converge to areal value. Hence, it is difficult to estimate the charge rate (SOC)accurately.

It is, hence, an object of the present invention to provide apparatusand method for estimating accurately the charge rate (SOC) for thesecondary cell and accurately estimating other parameters related to thecharge rate (SOC).

According to one aspect of the present invention, there is provided acharge rate estimating apparatus for a secondary cell, comprising: acurrent detecting section capable of measuring a current flowing throughthe secondary cell; a terminal voltage detecting section capable ofmeasuring a voltage across terminals of the secondary cell; a parameterestimating section that calculates an adaptive digital filtering using acell model in a continuous time series shown in an equation (1) andestimates all parameters at one time, the parameters corresponding to anopen-circuit voltage V₀, which is an offset term of the equation, (1)and coefficients of A(s), B(s), and C(s), which are transient terms; anda charge rate estimating section that estimates the charge rate from apreviously derived relationship between an open-circuit voltage and acharge rate of the secondary cell and the open-circuit voltage V₀,

$\begin{matrix}{{V = {{\frac{B(s)}{A(s)} \cdot I} + {\frac{1}{C(s)} \cdot V_{0}}}},} & (1)\end{matrix}$

wherein s denotes a Laplace transform operator, A(s), B(s), and C(s)denote poly-nominal functions of s.

According to another aspect of the present invention, there is provideda charge rate estimating method for a secondary cell, comprising:measuring a current flowing through the secondary cell; measuring avoltage across terminals of the secondary cell; calculating an adaptivedigital filtering using a cell model in a continuous time series shownin an equation (1); estimating all parameters at one time, theparameters corresponding to an open-circuit voltage V₀, which is anoffset term of the equation (1), and coefficients of A(s), B(s), andC(s), which are transient terms; and estimating the charge rate from apreviously derived relationship between an open-circuit voltage and acharge rate of secondary cell, and the open-circuit voltage V₀,

$\begin{matrix}{{V = {{\frac{B(s)}{A(s)} \cdot I} + {\frac{1}{C(s)} \cdot V_{0}}}},} & (1)\end{matrix}$

wherein s denotes a Laplace transform operator, A(s), B(s), and C(s)denote poly-nominal functions of s.

According to a still another object of the present invention, there isprovided a charge rate estimating method for a secondary cell,comprising: measuring a current I(k) flowing through the secondary cell;measuring a terminal voltage V(k) across the secondary cell; storing theterminal voltage V(k) when a current is zeroed as an initial value ofthe terminal voltage ΔV(k)=V(k)−V_ini; determining instantaneous currentvalues I₀(k), I₁(k), and I₃(k) and instantaneous terminal voltagesV₁(k), V₂(k), and V₃(k) from an equation (19),

$\begin{matrix}{{I_{0} = {\frac{1}{G_{1}(s)} \cdot I}},} & \; & \; & \; \\{{I_{1} = {\frac{s}{G_{1}(s)} \cdot I}},} & \; & \; & {{V_{1} = {\frac{s}{G_{1}(s)} \cdot V}},} \\{{I_{2} = {\frac{s^{2}}{G_{1}(s)} \cdot I}},} & \; & \; & {{V_{2} = {\frac{s^{2}}{G_{1}(s)} \cdot V}},} \\{{I_{3} = {\frac{s^{3}}{G_{1}(s)} \cdot I}},} & \; & \; & {{V_{3} = {\frac{s^{3}}{G_{1}(s)} \cdot V}},}\end{matrix}$

$\begin{matrix}{{\frac{1}{G_{1}(s)} = \frac{1}{\left( {{p\;{1 \cdot s}} + 1} \right)^{3}}},} & (19)\end{matrix}$

wherein p1 denotes a constant determining a responsive characteristic ofG₁(s); substituting the instantaneous current values I₀(k), I₁(k),I₂(k), and I₃(k) and the instantaneous terminal voltages V₁(k), V₂(k),and V₃(k) into an equation (18),

$\begin{matrix}{{\gamma(k)} = \frac{\lambda_{3}(k)}{1 + {{\lambda_{3}(k)} \cdot {\omega^{T}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\omega(k)}}}} & (18) \\{{\theta(k)} = {{\theta\left( {k - 1} \right)} - {{\gamma(k)} \cdot {P\left( {k - 1} \right)} \cdot {\omega(k)} \cdot \left\lbrack {{{\omega^{T}(k)} \cdot {\theta\left( {k - 1} \right)}} - {y(k)}} \right\rbrack}}} & \; \\{{P(k)} = {\frac{1}{\lambda_{1}(k)}\left\{ {{P\left( {k - 1} \right)} - \frac{{\lambda_{3}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\omega(k)} \cdot {\omega^{T}(k)} \cdot {P\left( {k - 1} \right)}}{1 + {{\lambda_{3}(k)} \cdot {\omega^{T}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\omega(k)}}}} \right\}}} & \; \\{\mspace{50mu}{= \frac{P^{\prime}(k)}{\lambda_{1}(k)}}} & \; \\{{\lambda_{1}(k)} = \left\{ {\frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{U}}:\mspace{14mu}{\lambda_{1} \leq \frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{U}}}} \right.} & \; \\{\mspace{85mu}\left\{ {\lambda_{1}:\mspace{14mu}{\frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{U}} \leq \lambda_{1} \leq \frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{L}}}} \right.} & \; \\{\mspace{85mu}\left\{ {{\frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{L}}:\mspace{14mu}{\frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{L}} \leq \lambda_{1}}},} \right.} & \;\end{matrix}$wherein θ(k) denotes a parameter estimated value at a time point of k(k=0, 1, 2, 3 - - - ), λ₁, λ₃(k), γu, and γL denote initial set value,b<λ₁<1, 0<λ₃(k)<∞. P(0) is a sufficiently large value, θ(0) provides aninitial value which is non-zero but very sufficiently small value,trace{P} means a trace of matrix P, wherein y(k)=V₁(k)

$\begin{matrix}{{\omega^{T}(k)} = \begin{bmatrix}{V_{3}(k)} & {V_{2}(k)} & {I_{3}(k)} & {I_{2}(k)} & {I_{1}(k)} & {I_{0}(k)}\end{bmatrix}} & (20) \\{{{\theta(k)} = \begin{bmatrix}{- {a(k)}} \\{- {b(k)}} \\{c(k)} \\{d(k)} \\{e(k)} \\{f(k)}\end{bmatrix}};} & \;\end{matrix}$substituting a, b, c, d, e, and f in the parameter estimated value θ(k)into and equation (22) to calculate V₀ which is an alternate of V₀ whichcorresponds to a variation ΔV₀(k) of the open-circuit voltage estimatedvalue from a time at which the estimated calculation start is carriedout;

$\begin{matrix}{{V_{0}^{\prime} = {{\frac{\left( {{T_{1} \cdot s} + 1} \right)}{G_{2}(s)} \cdot V_{0}} = {{a \cdot v_{6}} + {b \cdot v_{5}} + v_{4} - {c \cdot I_{6}} - {d \cdot I_{5}} - {e \cdot I_{4}}}}};} & (22)\end{matrix}$and calculating an open-circuit voltage estimated value V₀(k) accordingthe variation ΔV₀(k) of the open-circuit voltage estimated value and theterminal voltage initial value V_ini.

This summary of the invention does not necessarily describe allnecessary features so that the invention may also be a sub-combinationof these described features.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a functional block diagram of an apparatus for estimating acharge rate (SOC) of a secondary cell in a preferred embodimentaccording to the present invention.

FIG. 2 is a specific circuit block diagram of the apparatus forestimating the charge rate (SOC) of the secondary cell in the preferredembodiment according to the present invention.

FIG. 3 is a model view representing an equivalent circuit model of thesecondary cell.

FIG. 4 is a correlation map representing a correlation between anopen-circuit voltage and a charge rate (SOC).

FIG. 5 is an operational flowchart for explaining an operation of amicrocomputer of a battery controller of the charge rate estimatingapparatus in the first preferred embodiment shown in FIG. 1.

FIGS. 6A, 6B, 6C, 6D, 6E, 6F, 6G, 6H and 6I are characteristic graphsrepresenting results of simulations of current, voltages, and variousparameters in a case of the charge rate estimating apparatus in theembodiment shown in FIG. 1

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Reference will hereinafter be made to the drawings in order tofacilitate a better understanding of the present invention.

FIG. 1 shows a functional block diagram of charge rate estimatingapparatus in a first preferred embodiment according to the presentinvention. In FIG. 1, a reference numeral 1 denotes a parameterestimating section based on a cell model with an open-circuit voltageV₀(k) as an offset term. In addition, a reference numeral 2 denotes aopen-circuit voltage calculating section to calculate open-circuitvoltage V₀(k), and a reference numeral 3 denotes a charge rateestimating section that calculate the charge rate from the open-circuitvoltage. In addition, a reference numeral 4 denotes a current Imeasuring block to detect current I(k) which is charged and dischargedinto and from the cell, and a reference numeral 5 denotes s terminalvoltage of the cell to measure the terminal voltage V(k).

FIG. 2 shows a block diagram representing a specific structure of thecharge rate estimating apparatus in the first embodiment. In thisembodiment, a load such as a motor is driven with the secondary cell andthe charge rate estimating apparatus is mounted in a system to chargethe secondary cell with a regenerative power of the motor (load). InFIG. 2, a reference numeral 10 denotes a secondary cell (simply called,a cell), a reference numeral 20 denotes a load such as a DC motor, areference numeral 30 denotes a battery controller (electronic controlunit) to estimate the charge rate (charge state) of the cell having amicrocomputer including a ROM (Read Only Memory), a RAM (Random AccessMemory), a CPU (central Processing Unit), and Input/Output Interface andother electronic circuits. A reference numeral 40 denotes a currentmeter to detect a current which is charged into or discharged from thecell, a reference numeral 50 denotes a voltage meter to detect theterminal voltage of the cell, a reference numeral 60 denotes atemperature meter to detect a temperature of the cell. These meters areconnected to battery controller 30. Battery controller 30 corresponds toparts of parameter estimating section 1, an open-circuit voltage V₀(k)and a charge rate estimating section 3. Current meter 40 corresponds tocurrent I(k) measuring section and voltage meter 50 correspond toterminal voltage V(k) measuring section 5.

First, a “cell model” used in the first embodiment will be describedbelow. FIG. 3 is an equivalent circuit representing an equivalentcircuit model of the secondary cell. The equivalent circuit model of thesecondary cell can be represented by the following equation (7)(=equation (6)).

$\begin{matrix}{V = {{\frac{K \cdot \left( {{T_{2} \cdot s} + 1} \right)}{{T_{1} \cdot s} + 1} \cdot I} + {\frac{1}{{T_{3} \cdot s} + 1}{V_{0}.}}}} & (7)\end{matrix}$

In equation (7), a model input is a current I[A] (a positive valuerepresents a charge and a negative value represents a discharge), amodel output is a terminal voltage V[V], an open-circuit voltage is V₀,K denotes an internal resistance, T₁ through T₃ denote time constants(T₁≠T₂≠T₃, T₁<<T₃) and s denotes a Laplace transform operator.

In this model based on equation of (7) is a reduction model (firstorder) in which a positive pole and a negative pole are not speciallyseparated from each other. However, it is possible to represent acharge-discharge characteristic of an actual cell relatively easily.Equation (7), in equation (1) of V=B(s)/A(s)·I+1/C(s)·V₀ - - - (1),A(s)=T₁·s+1, B(s)=K·(T₂·s+1), C(s)=T₃·s+1.

Hereinafter, a deviation from the cell model based on equation (7) to anadaptive digital filter will first be described below. Open-circuitvoltage V₀ can be described by an equation (8), supposing that a valueof a current I multiplied with a variable efficiency of A is integratedfrom a certain initial state.

That is to say,

$\begin{matrix}{V_{0} = {\frac{A}{s} \cdot {I.}}} & (8)\end{matrix}$

It is noted that equation (8) corresponds to a replacement of h recitedin equation (2), viz., V₀=h/s·I with efficiency of A.

If equation (8) is substituted into equation (7), equation (9) isresulted.

$\begin{matrix}{V_{0} = {{\frac{K \cdot \left( {{T_{2} \cdot s} + 1} \right)}{{T_{1} \cdot s} + 1} \cdot I} + {\frac{1}{{T_{3} \cdot s} + 1} \cdot \frac{A}{s} \cdot {I.}}}} & (9)\end{matrix}$

Equation (9) corresponds to equation (3)

$\begin{pmatrix}{V = {{\left( {\frac{B(s)}{A(s)} + {\frac{1}{C(s)} \cdot \frac{h}{s}}} \right) \cdot I} = {\frac{{s \cdot {B(s)} \cdot {C(s)}} + {h \cdot {A(s)}}}{s \cdot {A(s)} \cdot {C(s)}} \cdot I}}} & \; & \; & (3)\end{pmatrix}.$A(s), B(s), and C(s) in equation (3), the following equations aresubstituted into equation (9) in the same way as the case of equation(7).

A(s)=T₁·s+1,

B(s)=K·(T₂·s+1)

C(s)=T₃·s+1. In other words, equation (3) is a generalized equation andthis application to a first order model is equation (9). If equation (9)is arranged, an equation of (10) is given.

$\begin{matrix}{{{{S \cdot \left( {{T_{1}s} + 1} \right)}{\left( {{T_{3} \cdot s} + 1} \right) \cdot V}} = {{{K \cdot \left( {{T_{2} \cdot s} + 1} \right)}\left( {{T_{3} \cdot s} + 1} \right){s \cdot I}} + {A \cdot \left( {{T_{1} \cdot s} + 1} \right) \cdot I}}}{{\left\{ {{T_{1} \cdot T_{3} \cdot s^{3}} + {\left( {T_{1} + T_{3}} \right) \cdot s^{2}} + s} \right\} \cdot V} = {{\left\{ {{K \cdot T_{2} \cdot T_{3} \cdot s^{3}} + {K \cdot \left( {T_{2} + T_{3}} \right) \cdot s^{2}} + {\left( {K + {A \cdot T_{1}}} \right) \cdot s} + A} \right\} \cdot {I\left( {{a \cdot s^{3}} + {b \cdot s^{2}} + s} \right)} \cdot V} = {\left( {{c \cdot s^{3}} + {d \cdot s^{2}} + {e \cdot s} + f} \right) \cdot {I.}}}}} & (10)\end{matrix}$It is noted that, in the last equation of equation (10), parameters arerewritten as follows:a=T ₁ ·T ₃ , b=T ₁ +T ₃ , c=K·T ₂ ·T ₃ , d=K·(T ₂ +T ₃), e=K+A·T ₁, andf=A  (11)If a stable low pass filter G₁(s) is introduced into both sides ofequation (10) and arranged, the following equation (12) is given.

$\begin{matrix}{{\frac{1}{G_{1}(s)}{\left( {{a \cdot s^{3}} + {b \cdot s^{2}} + s} \right) \cdot V}} = {\frac{1}{G_{1}(s)}{\left( {{c \cdot s^{3}} + {d \cdot s^{2}} + {e \cdot s} + f} \right) \cdot {I.}}}} & (12)\end{matrix}$In details, in equation (10), on the contrary of equation (7), ifT₁·s+1=A(s), K·(T₂·s+1)=B(s), and T₃·s+1=C(s) are substituted intoequation (10), this is given as: s·A(s)·C(s)·V=B(s)·C(s)·s·I+A·A(s)·I.This is rearranged as follows: s·A(s)·C(s)·V=[B(s)·C(s)·s[[·I]]+A·A(s)]·I - - - (12)′. If, the low pass filter (LPF), G₁(s) isintroduced into both sides of equation (12)′, an equation (4) is given.That is to say,

$\begin{matrix}{{\frac{s \cdot {A(s)} \cdot {C(s)}}{G_{1}(s)} \cdot V} = {\frac{{s \cdot {B(s)} \cdot {C(s)}} + {h \cdot {A(s)}}}{s \cdot {A(s)} \cdot {C(s)}} \cdot {I.}}} & (4)\end{matrix}$It is noted that s denotes the Laplace transform operator, A(s), B(s),and C(s) denote a poly-nominal function of s, h denotes a variable, and1/G₁(s) denotes a transfer function having a low pass filtercharacteristic. That is to say, equation (4) is the generalizedfunction, equation (12) is the application of equation (4) to the firstorder model.

Current I and terminal voltage V which can actually be measured areprocessed by means of a low pass filter (LPF) and a band pass filter(BPF) are defined in the following equations (13), provided that p₁denotes a constant to determine a responsive characteristic of G1(s) andis determined according to a designer's desire.

$\begin{matrix}\begin{matrix}{I_{0} = {\frac{1}{G_{1}(s)} \cdot I}} & \; & \; & \; \\{{I_{1} = {\frac{s}{G_{1}(s)} \cdot I}},} & \; & \; & {{V_{1} = {\frac{s}{G_{1}(s)} \cdot V}},} \\{{I_{2} = {\frac{s^{2}}{G_{1}(s)} \cdot I}},} & \; & \; & {{V_{2} = {\frac{s^{2}}{G_{1}(s)} \cdot V}},} \\{{I_{3} = {\frac{s^{3}}{G_{1}(s)} \cdot I}},} & \; & \; & {{V_{3} = {\frac{s^{3}}{G_{1}(s)} \cdot V}},} \\{\frac{1}{G_{1}(s)} = \frac{1}{\left( {{P_{1} \cdot s} + 1} \right)^{3}}} & \; & \; & \;\end{matrix} & (13)\end{matrix}$If equation (12) is rewritten using the variables shown in equations(13), equations (14) are represented and, if deformed, the followingequation (15) is given.a·V ₃ +b·v ₂ +v ₁ =c·I ₃ +d·I ₂ +e·I ₁+f·I₀V ₁ =−a·V ₃ −b·V ₂ +c·I ₃ +d·I ₂ +e·I ₁ +f·I ₀  (14)

$\begin{matrix}{V_{1} = {\begin{bmatrix}V_{3} & V_{2} & I_{3} & I_{2} & I_{1} & I_{0}\end{bmatrix} = {\begin{bmatrix}{- a} \\{- b} \\c \\d \\e \\f\end{bmatrix}.}}} & (15)\end{matrix}$Equation (15) is a product-sum equation of measurable values and unknownparameters. Hence, a standard (general) type (equation (16)) of theadaptive digital filter is coincident with equation (15). It is notedthat ω^(T) means a transposed vector in which a row and column of avector ω are mutually exchanged.

y=ω^(T)·θ - - - (16). It is noted that y, ω^(T), and θ can be expressedin the following equation (17) in equation (16) described above.

$\begin{matrix}{{Y = V_{1}},{\omega^{T} = \begin{bmatrix}V_{3} & V_{2} & I_{3} & I_{2} & I_{1} & I_{0}\end{bmatrix}},{\theta = {\begin{bmatrix}{- a} \\{- b} \\c \\d \\e \\f\end{bmatrix}.}}} & (17)\end{matrix}$Hence, if a signal filter processed for current I and terminal voltage Vis used in a digital filter process calculation, unknown parametervector θ can be estimated.

In this embodiment, ″ a both-limitation trace gain method is used whichimproves a logical demerit of a simple ″ an adaptive digital filter bymeans of a least square method ″ such that once the estimated value isconverged, an accurate estimation cannot be made any more even if theparameters are changed. A parameter estimating algorithm to estimateunknown parameter vector θ with equation (16) as a prerequisite is asshown in an equation (18). It is noted that the parameter estimatedvalue at a time point of k is θ(k).

$\begin{matrix}{{\gamma(k)} = \frac{\lambda_{3}(k)}{1 + {{\lambda_{3}(k)} \cdot {\omega^{T}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\omega(k)}}}} & (18) \\{{\theta(k)} = {{\theta\left( {k - 1} \right)} - {{\gamma(k)} \cdot {P\left( {k - 1} \right)} \cdot {\omega(k)} \cdot \left\lbrack {{{\omega^{T}(k)} \cdot {\theta\left( {k - 1} \right\rangle}} - {y(k)}} \right\rbrack}}} & \; \\{{P(k)} = {\frac{1}{\lambda_{1}(k)}\left\{ {{P\left( {k - 1} \right)} - \frac{{\lambda_{3}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\omega(k)} \cdot {\omega^{T}(k)} \cdot {P\left( {k - 1} \right)}}{1 + {{\lambda_{3}(k)} \cdot {\omega^{T}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\omega(k)}}}} \right\}}} & \; \\{\mspace{50mu}{= \frac{P^{\prime}(k)}{\lambda_{1}(k)}}} & \; \\{{\lambda_{1}(k)} = \left\{ {\frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{U}}:\mspace{14mu}{\lambda_{1} \leq \frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{U}}}} \right.} & \; \\{\mspace{85mu}\left\{ {\lambda_{1}:\mspace{14mu}{\frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{U}} \leq \lambda_{1} \leq \frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{L}}}} \right.} & \; \\{\mspace{85mu}\left\{ {\frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{L}}:\mspace{14mu}{\frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{L}} \leq {\lambda_{1}.}}} \right.} & \;\end{matrix}$

In equations (18), λ₁, λ₃(k), γ_(u), and γ_(L) denote initial set value,b<λ₁<1, 0<λ₃(k)<∞. P(0) is a sufficiently large value, θ(0) provides aninitial value which is non-zero but very sufficiently small value. Inaddition, trace{P} means a trace of matrix P. As described above, thederivation of the adaptive digital filter from cell model.

FIG. 5 shows an operational flowchart carrying out the microcomputer ofbattery controller 30. A routine shown in 5 is carried out for eachconstant period of time T₀. For example, I(k) is the present value andI(k−1) means a one previous value of I(k). At a step S10, batterycontroller 30 measures current I(k) and I(k−1) means one previous valueof I(k). At a step S20, battery controller 30 carries out a turnon-and-off determination of an interrupt relay of the secondary cell.That is to say, battery controller 30 performs the on-and-off control ofthe interrupt relay of the secondary cell. When a relay is turned off(current I=0), the routine goes to a step S30. During the engagement ofthe relay, the routine goes to a step S40. At step S30, when the relayis engaged, the routine goes to a step S540. At step S530, batterycontroller 30 serves to store terminal voltage V(k) to as an initialvalue of the terminal voltage V_ini. At a step S40, battery controller30 calculates a differential value ΔV(k) of the terminal voltage.ΔV(k)=V(k)−V_ini. This is because the initial value of the estimationparameter in the adaptive digital filter is 0 so that the estimationparameter does not converge during the estimation calculation starttime. Thus, all of inputs are zeroed. During the input being all zeroed.During the relay interruption, step S30 have been passed and theestimation parameters are remains initial state since I=0 and theestimation parameter remains alive.

At step S50, a low pass filtering or band pass filtering are carried outthe current I(k) and terminal voltage difference value ΔV(k) on thebasis of equation (13). I₀(k) through I₃(k) and V₁(k) through V₃(k) arecalculated from equation (19). In this case, in order to improve anestimation accuracy of the parameter estimation algorithm of equation(18), a responsive characteristic of low pass filter G₁(s) is set to beslow so as to reduce observation noises. However, if the characteristicis quicker than a response characteristic of the secondary cell (a roughvalue of time constant T₁ is known), each parameter of the electric cellmodel cannot accurately be estimated. It is noted that p₁ recited inequation (19) denotes a constant determined according to the responsivecharacteristic of G₁(s).

$\begin{matrix}\begin{matrix}{{I_{0} = {\frac{1}{G_{1}(s)} \cdot I}},} & \; & \; & \; \\{{I_{1} = {\frac{s}{G_{1}(s)} \cdot I}},} & \; & \; & {{V_{1} = {\frac{s}{G_{1}(s)} \cdot V}},} \\{{I_{2} = {\frac{s^{2}}{G_{1}(s)} \cdot I}},} & \; & \; & {{V_{2} = {\frac{s^{2}}{G_{1}(s)} \cdot V}},} \\{{I_{3} = {\frac{s^{3}}{G_{1}(s)} \cdot I}},} & \; & \; & {{V_{3} = {\frac{s^{3}}{G_{1}(s)} \cdot V}},{and}} \\{\frac{1}{G_{1}(s)} = {\frac{1}{\left( {{p\;{1 \cdot s}} + 1} \right)^{3}}.}} & \; & \; & \;\end{matrix} & (19)\end{matrix}$At a step S60, I₀(k) through I₃(k) calculated at step S50 and V₁(k)through V₃(k) are substituted into equation (18). Then, the parameterestimation algorithm in the adaptive digital filter, viz., equation (18)is executed to calculate parameter estimated value θ(k). y(k), ω^(T)(k),and θ(k) are shown in equation (20).

$\begin{matrix}\begin{matrix}{{y(k)} = {V_{1}(k)}} \\{{\omega^{T}(k)} = \left\lbrack {{V_{3}(k)}\mspace{31mu}{V_{2}(k)}\mspace{31mu}{I_{3}(k)}\mspace{31mu}{I_{2}(k)}\mspace{31mu}{I_{1}(k)}\mspace{31mu}{I_{0}(k)}} \right\rbrack} \\{{\theta(k)} = {\begin{bmatrix}{- {a(k)}} \\{- {b(k)}} \\{c(k)} \\{d(k)} \\{e(k)} \\{f(k)}\end{bmatrix}.}}\end{matrix} & (20)\end{matrix}$At a step S70, a through e of parameter estimated value θ(k) calculatedat step S60 are substituted into the following equation (22) in whichthe above-described cell model equation (7) is deformed to calculate V₀′which is an alternative to open-circuit voltage V₀. Since the variationin open-circuit voltage V₀ is smooth, V₀′ can be used alternatively. Itis noted that the derivation herein is a variation ΔV₀(k) of theopen-circuit voltage from the estimated calculation start time.

It is noted that an equation of [1/C1(s)]I in equation (21) is replacedwith an equation (24) corresponds to equation (22). It is also notedthat, in the derivation of equation (22), K in equation (21) is strictlydifferent from e in equation (21). However, since, physically, K>>A·T₁,e is approximated to K(e≅K). Then, each coefficient a through e inequation (22) is the contents shown in equation (23).

$\begin{matrix}\begin{matrix}{{\frac{1}{{T_{3} \cdot s} + 1} \cdot V_{0}} = {V - {\frac{K \cdot \left( {T_{2} + s + 1} \right)}{{T_{1} \cdot s} + 1} \cdot I}}} \\{{\left( {{T_{1} \cdot s} + 1} \right) \cdot {Vo}} = {{\left( {{T_{1} \cdot s} + 1} \right)\left( {{T_{3} \cdot s} + 1} \right)V} - {{K \cdot \left( {{T_{2} \cdot s} + 1} \right)}{\left( {{T_{3} \cdot s} + 1} \right) \cdot I}}}} \\{{\left( {{T_{1} \cdot s} + 1} \right) \cdot V_{0}} = {{\left\{ {{T_{1} \cdot T_{3} \cdot s^{2}} + {\left( {T_{1} + T_{3}} \right) \cdot s} + 1} \right\} \cdot V} - {\left\{ {{K \cdot T_{2} \cdot T_{3} \cdot s^{2}} + {K \cdot \left( {T_{2} + T_{3}} \right) \cdot s} + K} \right\} \cdot I}}} \\{{\frac{\left( {{T_{1} \cdot s} + 1} \right)}{G_{2}} \cdot V_{0}} = {\frac{1}{G_{2}(s)}{\left( {{a \cdot s^{2}} + {b \cdot s} + K} \right) \cdot {I.}}}}\end{matrix} & (21) \\{V_{0}^{\prime} = {{\frac{\left( {{T_{1} \cdot s} + 1} \right)}{G_{2}(s)} \cdot V_{0}} = {{a \cdot V_{6}} + {b \cdot V_{5}} + V_{4} - {c \cdot I_{6}} - {d \cdot I_{5}} - {e \cdot {I_{4}.}}}}} & (22)\end{matrix}$

It is noted that a=T₁·T₃, b=T₁+T₃, c=K·(T₂+T₃}, d=K·(T₂+T₃),e=K+A·T₁=K - - - (23).

$\begin{matrix}\begin{matrix}{{I_{4} = {\frac{1}{G_{2}(s)} \cdot I}},} & {{V_{4} = {\frac{1}{G_{2}(s)} \cdot V}},} \\{{I_{5} = {\frac{s}{G_{2}(s)} \cdot I}},} & {{V_{5} = {\frac{s}{G_{2}(s)} \cdot V}},} \\{{\frac{1}{G_{2}(s)} = {\frac{1}{{p_{2} \cdot s} + 1} \cdot \frac{1}{{T_{1}^{\prime} \cdot s} + 1}}},} & \; \\{{I_{6} = {\frac{s^{2}}{G_{2}(s)} \cdot I}},{{{and}\mspace{14mu} V_{6}} = {\frac{s^{2}}{G_{2}(s)} \cdot {V.}}}} & \;\end{matrix} & (24)\end{matrix}$

p₂ recited in equations (24) denote a constant to determine a responsivecharacteristic of G₂(s). T₁ of the cell parameter is known to be severalseconds. Hence, T′₁ in equation (24) is set to be approximated value toT₁. Thereby, since (T₁·s+1) which remains in a numerator of equation(22) can be compensated, the estimation accuracy of open-circuit voltageV₀ can be improved. It is noted that equation (21) corresponds toequation (5). That is to say, equation (21) is derived from(T₁·s+1)·V₀=(T₁·s+1)(T₃·S+1)·V−K·(T₂·s+1)(T₃·s+1)·(T₃·s+1)·I. If thefollowing three equations are substituted into the above-describeddeformation of equation (21). T₁·s+1=A(s), K·(T₂·s+1) B(s), andT₃·s+1=C(s). That is to say, A(s)·V₀=A(s)·C(s)·V−B(s)·C(s)·I. If this isrearranged, this results in V₀=C(s)·V−B(s)·C(s)·I/A(s),V₀=C(s)·[V−B(s)·I/A(s)] If low pass filter G₂(s) is introduced into bothsides of this equation, this results in equation (5). In details,equation (5) is a generalization equation and the application ofequation (5) to the first order model is equation (2).

At a step S80, battery controller 30 adds the open-circuit voltageinitial value, i.e., terminal voltage initial value V_ini to a variationΔV₀(k) of open-circuit voltage V₀ so as to obtain open-circuit voltageestimated value V₀(k) from the following equation (25).V ₀(k)=ΔV ₀(k)+V _(—) ini  (25).

At a step S90, battery controller 30 calculates the charge rate SOC(k)from open-circuit voltage V₀(k) calculated at step S80 using acorrelation map of the open-circuit voltage versus the charge rate asshown in FIG. 4. It is noted that, in FIG. 4, V_(L) denotes theopen-circuit voltage corresponding to SOC=0% and V_(H) denotes theopen-circuit voltage corresponding to SOC=100%. At a step S100, batterycontroller 30 stores the necessary numerical values needed in thesubsequent calculation and the present routine is ended. As describedabove, an operation of the apparatus for estimating the charge rate ofthe secondary cell has been described.

(1) As described above, a relationship from among current I of thesecondary cell and terminal voltage V thereof, and the open-circuitvoltage V₀ is structured in transfer function that as in the generalequation (1), that in the preferred embodiment, equation (7) (=equation(6). Hence, it is made possible to apply an adaptive digital filter suchas a least square method (well known estimation algorithm).Consequently, it becomes possible to estimate parameters in equations(viz., open-circuit voltage V₀ which is an offset term and poly-nominalequations A(s), B(s), and C(s)) in a form of a batch processing. Theseparameters are largely affected by the charge rate, a surroundingtemperature, and a deterioration and varied instantaneously. It ispossible to sequentially estimate the adaptive digital filter with goodaccuracy. Then, if a unique correlation between the open-circuit voltageV₀ and the charge rate as shown in FIG. 4 are stored, the estimatedopen-circuit voltage can be converted to the charge rate. Hence, it ispossible to sequentially estimate the charge rate in the same way as theparameters described above.

(2) In a case where the equation (1) which is the relationship equationof current I and terminal voltage V of the secondary cell isapproximated to equation (4), the equation such that no offset term isincluded (viz., the open-circuit voltage V₀), a product-and-additionequation between a measurable current I which is filter processed and aterminal voltage V which is filter processed and unknown parameter(coefficient parameters of poly-nominal equations A(s), B(s), and C(s)and h) is obtained. A normally available adaptive digital filter (theleast mean square method and well known parameter estimation algorithm)can directly be applied in a continuous time series.

As a result of this, the unknown parameters can be estimated in thebatch processing manner and the estimated parameter h is substitutedinto equation (2), the estimated value of open-circuit voltage V₀ caneasily be calculated. All of these parameters are variedinstantaneously, the adaptive digital filter can serve to estimate thecharge rate at any time with a high accuracy. Since a constantrelationship between open-circuit voltage V₀ and the charge rate (SOC)is established as shown in FIG. 4, if this relationship is previouslystored, the charge rate (SOC) can be estimated from the estimated valueof open-circuit voltage V₀.

FIGS. 6A through 6I integrally shows signal timing charts with current Iand terminal voltage V inputted into adaptive digital filter andrepresenting results of simulation graphs when each parameter isestimated. As far as a time constant of a first order delay in equation(6) is concerned, T₁<T₀. Since all parameters a through f (refer toequation (11)) are favorably estimated, the estimated value ofopen-circuit voltage V₀ can be said to be well coincident with a realvalue.

It is noted that, in FIG. 6C which indicates the open-circuit voltage, areason that a right side second term of equation (6) is described is toindicate that the open-circuit voltage estimated value is coincidentwith a real value almost without delay in spite of the fact that a lateterm of time constant T₃ is measured on the terminal voltage inputtedinto the adaptive filter. In details, since the parameter estimationwith the cell model formatted adaptive digital filter in equation (6),all of parameters a through f can favorably be estimated and theestimated value of open-circuit voltage V₀ is well coincident with areal value.

(3) In addition, as described in item (2), in the structure in which theopen-circuit voltage V₀ is calculated from equation (2), the integrationoccurs before a value at which estimated value h is converged to thereal value, its error cannot be eliminated. However, in the structure inwhich equation (5) in which the integration is not included, the errorbefore the parameter estimated value is converged into the real valuedoes not give an influence after the convergence.

It will be appreciated that, in part of {circle around (1)} in FIG. 6I,before estimated value f is converged into a real value, an erroneousestimation is carried out only at momentarily. In equation (2), thisvalue is also integrated so that the error is not eliminated. However,the error is not eliminated since even this value is integrated.However, in the structure using equation (5), open-circuit voltage V₀ iscalculated from the equation in which the integration is not included.Hence, after the parameter estimated value is converged into the realtime, this erroneous estimation portion is eliminated.

(4) Furthermore, in a case where equation (6) is used in place ofequation (1), a calculation time and program capacity can be suppressedto a minimum while having the above-described advantages.

The entire contents of a Japanese Patent Application No. 2002-340803(filed in Japan on Nov. 25, 2002) are herein incorporated by reference.The scope of the invention is defined with reference to the followingclaims.

1. A charge rate estimating apparatus for a secondary cell, comprising:a current detecting section capable of measuring a current flowingthrough the secondary cell; a terminal voltage detecting section capableof measuring a voltage across terminals of the secondary cell; aparameter estimating section that calculates an adaptive digitalfiltering using a cell model in a continuous time series shown in anequation (1) and estimates all parameters at one time, the parameterscorresponding to an open-circuit voltage V₀ which is an offset term ofthe equation (1) and coefficients of A(s), B(s), and C(s), which aretransient terms; and a charge rate estimating section that estimates thecharge rate from a previously derived relationship between anopen-circuit voltage and a charge rate of the secondary cell and theopen-circuit voltage V₀, $\begin{matrix}{{V = {{\frac{B(s)}{A(s)} \cdot I} + {\frac{1}{C(s)} \cdot V_{0}}}},} & (1)\end{matrix}$ wherein s denotes a Laplace transform operator, A(s),B(s), and C(s) denote poly-nominal functions of s, wherein theopen-circuit voltage V₀ of the cell model in the continuous time seriesshown in the equation (1) is approximated by means of an equation (2) toprovide an equation (3) and the adaptive digital filter calculation iscarried out using the equation (3) and equivalent equation (4), h isestimated in at least equation (4), the estimated h is substituted intoequation (2) to derive an open-circuit voltage V₀, and the charge rateis estimated from the previously derived relationship between anopen-circuit voltage and a charge rate of the secondary cell and theopen-circuit voltage V₀ $\begin{matrix}{V_{0} = {\frac{h}{s} \cdot I}} & (2) \\{V = {{\left( {\frac{B(s)}{A(s)} + {\frac{1}{C(s)} \cdot \frac{h}{s}}} \right) \cdot I} = {\frac{{s \cdot {B(s)} \cdot {C(s)}} + {h \cdot {A(s)}}}{s \cdot {A(s)} \cdot {C(s)}} \cdot I}}} & (3) \\{{{\frac{s \cdot {A(s)} \cdot {C(s)}}{G_{1}(s)} \cdot V} = {\frac{{s \cdot {B(s)} \cdot {C(s)}} + {h \cdot {A(s)}}}{G_{1}(s)} \cdot I}},} & (4)\end{matrix}$ wherein s denotes the Laplace transform operator, A(s),B(s), and C(s) denote poly-nominal equation functions, h denotes avariable, and 1/G₁(s) denotes a transfer function having a low passfilter characteristic.
 2. A charge rate estimating apparatus for asecondary cell, comprising: a current detecting section capable ofmeasuring a current flowing through the secondary cell; a terminalvoltage detecting section capable of measuring a voltage acrossterminals of the secondary cell; a parameter estimating section thatcalculates an adaptive digital filtering using a cell model in acontinuous time series shown in an equation (1) and estimates allparameters at one time, the parameters corresponding to an open-circuitvoltage V₀, which is an offset term of the equation (1), andcoefficients of A(s), B(s), and C(s), which are transient terms; and acharge rate estimating section that estimates the charge rate from apreviously derived relationship between an open-circuit voltage and acharge rate of the secondary cell and the open-circuit voltage V₀,$\begin{matrix}{{V = {{\frac{B(s)}{A(s)} \cdot I} + {\frac{1}{C(s)} \cdot V_{0}}}},} & (1)\end{matrix}$ wherein s denotes a Laplace transform operator, A(s),B(s), and C(s) denote poly-nominal functions of s, wherein theopen-circuit voltage V₀ of the cell model in the time continuous timeseries is approximated in an equation (2) to calculate an equation (3),the adaptive digital filter calculation is carried out using an equation(4) which is equivalent to the equation (3), A(s), B(s), and C(s) areestimated from equation (4), the estimated A(s), B(s), and C(s) aresubstituted into equation (5) to determine V₀/G₂(s) and the charge rateis estimated from the previously derived relationship between anopen-circuit voltage and a charge rate of the secondary cell and theopen-circuit voltage V₀ using the derived V₀/G₂(s) in place of theopen-circuit voltage V₀, $\begin{matrix}{V_{0} = {\frac{h}{s} \cdot I}} & (2) \\{V = {{\left( {\frac{B(s)}{A(s)} + {\frac{1}{C(s)} \cdot \frac{h}{s}}} \right) \cdot I} = {\frac{{s \cdot {B(s)} \cdot {C(s)}} + {h \cdot {A(s)}}}{s \cdot {A(s)} \cdot {C(s)}} \cdot I}}} & (3) \\{{{\frac{s \cdot {A(s)} \cdot {C(s)}}{G_{1}(s)} \cdot V} = {\frac{{s \cdot {B(s)} \cdot {C(s)}} + {h \cdot {A(s)}}}{G_{1}(s)} \cdot I}},} & (4) \\{{\frac{V_{0}}{G_{2}(s)} = {\frac{C(s)}{G_{2}(s)} \cdot \left( {V - {\frac{B(s)}{A(s)} \cdot I}} \right)}},} & (5)\end{matrix}$ wherein s denotes the Laplace transform operator, A(s),B(s), and C(s) denote the polynominal (equation) function of s, hdenotes a variable, 1/G₁(s) and 1/G₂(s) denote transfer functions havingthe low pass filter characteristics.
 3. A charge rate estimatingapparatus for a secondary cell as claimed in claim 1, wherein the cellmodel is calculated from an equation (6),${V = {{\frac{K \cdot \left( {{T_{2} \cdot s} + 1} \right)}{{T_{1} \cdot s} + 1} \cdot I} + {\frac{1}{{T_{3} \cdot s} + 1}V_{0}}}},$wherein K denotes an internal resistance of the secondary cell, T₁, T₂,and T₃ denote time constants and, 1/G₁(s) denotes a low pass filterhaving a third order or more, wherein A(s)=T₁·s+1, B(s)=K·(T₂ S+1),C(s)=T₃·S+1.
 4. A charge rate estimating apparatus for a secondary cellas claimed in claim 3, wherein $\begin{matrix}{V = {{\frac{K \cdot \left( {{T_{2} \cdot s} + 1} \right)}{{T_{1} \cdot s} + 1} \cdot I} + {\frac{1}{{T_{3} \cdot s} + 1} \cdot \frac{A}{s} \cdot I}}} & (9)\end{matrix}$ (a·s³+b·s²+s)·V=(c·s³+d·s²+e·s+f)·I - - - (10).
 5. Acharge rate estimating apparatus for a secondary cell as claimed inclaim 4, wherein a=T₁·T₃, b=T₁+T₃, c=K·T₂·T₃, d=K·(T₂+T₃), e=K+A·T₁,f=A - - - (11).
 6. A charge rate estimating apparatus for a secondarycell as claimed in claim 5, wherein a stable low pass filter G₁(s) isintroduced into both sides of the equation (10) to derive the followingequation: $\begin{matrix}{{\frac{1}{G_{1}(s)}{\left( {{a \cdot s^{3}} + {b \cdot s^{2}} + s} \right) \cdot V}} = {\frac{1}{G_{1}(s)}{\left( {{c \cdot s^{3}} + {d \cdot s^{2}} + {e \cdot s} + f} \right) \cdot {I.}}}} & (12)\end{matrix}$
 7. A charge rate estimating apparatus for a secondary cellas claimed in claim 6, wherein actually measurable currents I andterminal voltages V which are processed by means of a low pass filterare as follows: $\begin{matrix}\begin{matrix}{{I_{0} = {\frac{1}{G_{1}(s)} \cdot I}},} & \; \\{{I_{1} = {\frac{s}{G_{1}(s)} \cdot I}},} & {{V_{1} = {\frac{s}{G_{1}(s)} \cdot V}},} \\{{I_{2} = {\frac{s^{2}}{G_{1}(s)} \cdot I}},} & {{V_{2} = {\frac{s^{2}}{G_{1}(s)} \cdot V}},} \\{{I_{3} = {\frac{s^{3}}{G_{1}(s)} \cdot I}},} & {{V_{3} = {\frac{s^{3}}{G_{1}(s)} \cdot V}},{and}} \\{\frac{1}{G_{1}(s)} = {\frac{1}{\left( {{{p1} \cdot s} + 1} \right)^{3}}.}} & \;\end{matrix} & (13)\end{matrix}$
 8. A charge rate estimating apparatus for a secondary cellas claimed in claim 7, wherein, using the equation (13), the equation of(12) is rewritten and rearranged as follows: $\begin{matrix}{V_{1} = {\begin{bmatrix}V_{3} & V_{2} & I_{3} & I_{2} & I_{1} & I_{0}\end{bmatrix} = \begin{bmatrix}{- a} \\{- b} \\c \\d \\e \\f\end{bmatrix}}} & (15)\end{matrix}$ and the equation (15) corresponds to a general equationwhich is coincident with a standard form of a general adaptive digitalfilter of equation (16): y=ω^(T)·θ - - - (16), wherein y=V₁, ω^(T)=[V₃V₂ I₃ I₂ I₁ I₀], and $\begin{matrix}{\theta = {\begin{bmatrix}{- a} \\{- b} \\c \\d \\e \\f\end{bmatrix}.}} & (17)\end{matrix}$
 9. A charge rate estimating apparatus for a Secondary cellas claimed in claim 8, wherein a parameter estimating algorithm with theequation (16) as a prerequisite is defined as follows: $\begin{matrix}{{\gamma(k)} = \frac{\lambda_{3}(k)}{1 + {{\lambda_{3}(k)} \cdot {\omega^{T}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\omega(k)}}}} & (18) \\{{\theta(k)} = {{\theta\left( {k - 1} \right)} - {{\gamma(k)} \cdot {P\left( {k - 1} \right)} \cdot {\omega(k)} \cdot \left\lbrack {{{\omega^{T}(k)} \cdot {\theta\left( {k - 1} \right)}} - {y(k)}} \right\rbrack}}} & \; \\{{P(k)} = {\frac{1}{\lambda_{1}(k)}\left\{ {{P\left( {k - 1} \right)} - \frac{{\lambda_{3}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\omega(k)} \cdot {\omega^{T}(k)} \cdot {P\left( {k - 1} \right)}}{1 + {{\lambda_{3}(k)} \cdot {\omega^{T}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\omega(k)}}}} \right\}}} & \; \\{\mspace{50mu}{= \frac{P^{\prime}(k)}{\lambda_{1}(k)}}} & \; \\{{\lambda_{1}(k)} = \left\{ {\frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{U}}:\mspace{14mu}{\lambda_{1} \leq \frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{U}}}} \right.} & \; \\{\mspace{85mu}\left\{ {\lambda_{1}:\mspace{14mu}{\frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{U}} \leq \lambda_{1} \leq \frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{L}}}} \right.} & \; \\{\mspace{85mu}\left\{ {{\frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{L}}:\mspace{14mu}{\frac{{trace}\left\{ {P^{\prime}(k)} \right\}}{\gamma_{L}} \leq \lambda_{1}}},} \right.} & \;\end{matrix}$ wherein θ(k) denotes a parameter estimated value at a timepoint of k (k=0, 1, 2, 3 - - - ), λ₁, λ₃(k), γu, and γL denote initialset value, b<λ₁<1, 0<λ₃(k)<∞, P(0) is a value, of P(k) at k=0, θ(0) isθ(k) at k=0 and θ(0) provides an initial value which is non-zero, andtrace{P} means a trace of matrix P.
 10. A charge rate estimating methodfor a secondary cell, comprising: measuring a current flowing throughthe secondary cell; measuring a voltage across terminals of thesecondary cell; calculating an adaptive digital filtering to provide anadaptive digital filter calculation using a cell model in a continuoustime series shown in an equation (1); estimating all parameters at onetime, the parameters corresponding to an open-circuit voltage V₀, whichis an offset term of the equation (1), and coefficients of A(s), B(s),and C(s), which are transient terms; and estimating the charge rate froma previously derived relationship between an open-circuit voltage and acharge rate of the secondary cell and the open-circuit voltage V₀,$\begin{matrix}{{V = {{\frac{B(s)}{A(s)} \cdot I} + {\frac{1}{C(s)} \cdot V_{0}}}},} & (1)\end{matrix}$ wherein s denotes a Laplace A(s) C(s) transform operator,A(s), B(s), and C(s) denote poly-nominal functions of s, wherein theopen-circuit voltage V₀ of the cell model in the continuous time seriesshown in the equation (1) is approximated by means of an equation (2) toprovide an equation (3) and the adaptive digital filter calculation iscarried out using the equation (3) and equivalent equation (4), h isestimated in at least equation (4), the estimated value of h issubstituted into equation (2) to derive an open-circuit voltage V₀, andthe charge rate is estimated from the previously derived relationshipbetween an open-circuit voltage and a charge rate of the secondary celland the open-circuit voltage V₀ and the charge rate $\begin{matrix}{V_{0} = {\frac{h}{s} \cdot I}} & (2) \\{V = {{\left( {\frac{B(s)}{A(s)} + {\frac{1}{C(s)} \cdot \frac{h}{s}}} \right) \cdot I} = {\frac{{s \cdot {B(s)} \cdot {C(s)}} + {h \cdot {A(s)}}}{s \cdot {A(s)} \cdot {C(s)}} \cdot I}}} & (3) \\{{{\frac{s \cdot {A(s)} \cdot {C(s)}}{G_{1}(s)} \cdot V} = {\frac{{s \cdot {B(s)} \cdot {C(s)}} + {h \cdot {A(s)}}}{G_{1}(s)} \cdot I}},} & (4)\end{matrix}$ wherein s denotes the Laplace transform operator, A(s),B(s), and C(s) denote poly-nominal equation functions, h denotes avariable, and 1/G₁(s) denotes a transfer function having a low passfilter characteristic.
 11. A charge rate estimating method for asecondary cell, comprising: measuring a current flowing through thesecondary cell; measuring a voltage across terminals of the secondarycell; calculating an adaptive digital filtering to provide an adaptivedigital filter calculation using a cell model in a continuous timeseries shown in an equation (1); estimating all parameters at one time,the parameters corresponding to an open-circuit voltage V₀, which is anoffset term of the equation (1), and coefficients of A(s), B(s), andC(s), which are transient terms; and estimating the charge rate from apreviously derived relationship between an open-circuit voltage and acharge rate of the secondary cell and the open-circuit voltage V₀,$\begin{matrix}{{V = {{\frac{B(s)}{A(s)} \cdot I} + {\frac{1}{C(s)} \cdot V_{0}}}},} & (1)\end{matrix}$ wherein s denotes a Laplace transform operator, A(s),B(s), and C(s) denote poly-nominal functions of s, wherein theopen-circuit voltage V₀ of the cell model in the continuous time seriesis approximated in an equation (2) to calculate an equation (3), theadaptive digital filter calculation is carried out using an equation (4)which is equivalent to the equation (3), A(s), B(s), and C(s) areestimated from the equation (4), the estimated A(s), B(s), and C(s) aresubstituted into equation (5) to determine V₀/G₂(s) and the charge rateis estimated from the previously derived relationship between anopen-circuit voltage and a charge rate of the secondary cell, and theopen-circuit voltage V₀ and the charge rate using the derived V₀/G₂(s)in place of the open-circuit voltage V₀, $\begin{matrix}{V_{0} = {\frac{h}{s} \cdot I}} & (2) \\{V = {{\left( {\frac{B(s)}{A(s)} + {\frac{1}{C(s)} \cdot \frac{h}{s}}} \right) \cdot I} = {\frac{{s \cdot {B(s)} \cdot {C(s)}} + {h \cdot {A(s)}}}{s \cdot {A(s)} \cdot {C(s)}} \cdot I}}} & (3) \\{{{\frac{s \cdot {A(s)} \cdot {C(s)}}{G_{1}(s)} \cdot V} = {\frac{{s \cdot {B(s)} \cdot {C(s)}} + {h \cdot {A(s)}}}{G_{1}(s)} \cdot I}},} & (4) \\{{\frac{V_{0}}{G_{2}(s)} = {\frac{C(s)}{G_{2}(s)} \cdot \left( {V - {\frac{B(s)}{A(s)} \cdot I}} \right)}},} & (5)\end{matrix}$ wherein s denotes the Laplace G₂ (s) G₂ (s) A(s) transformoperator, A(s), B(s), and C(s) denote the polynominal (equation)function of s, h denotes a variable, 1/G₁(s) and 1/G₂(s) denote transferfunctions having the low pass filter characteristics.
 12. A charge rateestimating method for a secondary cell as claimed in claim 10, whereinthe cell model is calculated from an equation (6),${V = {{\frac{K \cdot \left( {{T_{2} \cdot s} + 1} \right)}{{T_{1} \cdot s} + 1} \cdot I} + {\frac{1}{{T_{3} \cdot s} + 1}V_{0}}}},$wherein K denotes an internal resistance of the secondary cell, T₁, T₂,and T₃ denote time constants, 1/G₁(s) denotes a low pass filter having athird order or more.
 13. A charge rate estimating apparatus for asecondary cell as claimed in claim 2, wherein the cell model iscalculated from an equation (6),$v = {{\frac{K \cdot \left( {{T_{2} \cdot s} + 1} \right)}{{T_{1} \cdot s} + 1} \cdot I} + {\frac{1}{{T_{3} \cdot s} + 1}v_{0}}}$wherein K denotes an internal resistance of the secondary cell, T₁, T₂,and T₃ denote time constants, 1/G₁(s) denotes a low pass filter having athird order or more, and 1/G₂(s) denotes another low pass filter havinga second order or more, wherein A(s)=T₁·s+1, B(s)=K·(T₂ S+1),C(s)=T₃·S+1.
 14. A charge rate estimating apparatus for a secondarycell, comprising: a current detecting section capable of measuring acurrent flowing through the secondary cell; a terminal voltage detectingsection capable of measuring a voltage across terminals of the secondarycell; a parameter estimating section that calculates an adaptive digitalfiltering using a cell model in a continuous time series shown in anequation (1) estimates all of parameters at one time, the parameterscorresponding to an open-circuit voltage V₀ which is an offset term ofthe equation (1) and coefficients of A(s), B(s), and C(s) which aretransient terms; and a charge rate estimating section that estimates thecharge rate from a previously derived relationship between anopen-circuit voltage and a charge rate of the secondary cell, and theopen-circuit voltage V₀, $\begin{matrix}{{V = {{\frac{B(s)}{A(s)} \cdot I} + {\frac{1}{C(s)} \cdot V_{0}}}},} & (1)\end{matrix}$ wherein s denotes a Laplace transform operator, A(s),B(s), and C(s) denote poly-nominal functions of s, Wherein theopen-circuit voltage V₀ of the cell model in the continuous time seriesshown in the equation (1) is approximated by means of an equation (2) toprovide an equation (3) and the adaptive digital filter calculation iscarried out using the equation (3) and equivalent equation (4), h isestimated in at least equation (4), the estimated h is substituted intoequation (2) to derive an open-circuit voltage V₀, and the charge rateis estimated from the previously derived relationship between anopen-circuit voltage and a charge rate of the secondary cell and theopen-circuit voltage V₀, $\begin{matrix}{v_{0} = {\frac{h}{s} \cdot I}} & (2)\end{matrix}$ Where equations (3) and (4) do not include theopen-circuit voltage V₀, but are product and addition equations relatingV and I.
 15. A charge rate estimating method for a secondary cell asclaimed in claim 11, wherein the cell model is calculated from anequation (6),$v = {{\frac{K \cdot \left( {{T_{2} \cdot s} + 1} \right)}{{T_{1} \cdot s} + 1} \cdot I} + {\frac{1}{{T_{3} \cdot s} + 1}v_{0}}}$wherein K denotes an internal resistance of the secondary cell, T₁, T₂,and T₃ denote time constants, 1/G₁(s) denotes a low pass filter having athird order or more, and 1/G₂(s) denotes another low pass filter havinga second order or more.